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Precessional instability of a fluid cylinder

Published online by Cambridge University Press:  06 January 2011

ROMAIN LAGRANGE ,
PATRICE MEUNIER ,
FRANÇOIS NADAL  and
CHRISTOPHE ELOY
ROMAIN LAGRANGE*
Affiliation:
IRPHE, CNRS, Aix-Marseille Université, 49 rue Joliot-Curie, 13013 Marseille, France
PATRICE MEUNIER
Affiliation:
IRPHE, CNRS, Aix-Marseille Université, 49 rue Joliot-Curie, 13013 Marseille, France
FRANÇOIS NADAL
Affiliation:
Commissariat à l'Energie Atomique, CESTA, 33114 le Barp, France
CHRISTOPHE ELOY
Affiliation:
IRPHE, CNRS, Aix-Marseille Université, 49 rue Joliot-Curie, 13013 Marseille, France
*
†Email address for correspondence: [email protected]
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Abstract

In this paper, the instability of a fluid inside a precessing cylinder is addressed theoretically and experimentally. The precessional motion forces Kelvin modes in the cylinder, which can become resonant for given precessional frequencies and cylinder aspect ratios. When the Reynolds number is large enough, these forced resonant Kelvin modes eventually become unstable. A linear stability analysis based on a triadic resonance between a forced Kelvin mode and two additional free Kelvin modes is carried out. This analysis allows us to predict the spatial structure of the instability and its threshold. These predictions are compared to the vorticity field measured by particle image velocimetry with an excellent agreement. When the Reynolds number is further increased, nonlinear effects appear. A weakly nonlinear theory is developed semi-empirically by introducing a geostrophic mode, which is triggered by the nonlinear interaction of a free Kelvin mode with itself in the presence of viscosity. Amplitude equations are obtained coupling the forced Kelvin mode, the two free Kelvin modes and the geostrophic mode. They show that the instability saturates to a fixed point just above threshold. Increasing the Reynolds number leads to a transition from a steady saturated regime to an intermittent flow in good agreement with experiments. Surprisingly, this weakly nonlinear model still gives a correct estimate of the mean flow inside the cylinder even far from the threshold when the flow is turbulent.

JFM classification

Instability: Instability Vortex Flows: Vortex Flows
Type
Papers
Information
Journal of Fluid Mechanics , Volume 666 , 10 January 2011 , pp. 104 - 145
Copyright
Copyright © Cambridge University Press 2011

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References

REFERENCES

Agrawal, B. N. 1993 Dynamics characteristics of liquid motion in partially filled tanks of a spinning spacecraft. J. Guid. Control Dyn. 16 (4), 636–640.CrossRefGoogle Scholar
Bao, G. W. & Pascal, M. 1997 Stability of a spinning liquid filled spacecraft. Appl. Mech. 67, 407–421.Google Scholar
Bayly, B. J. 1986 Three-dimensional instability of elliptical flow. Phys. Rev. Lett. 57, 2160–2163.CrossRefGoogle ScholarPubMed
Busse, F. 1968 Steady fluid flow in a precessing spheroidal shell. J. Fluid Mech. 33, 739–752.CrossRefGoogle Scholar
Eloy, C., Le Gal, P. & Le Dizès, S. 2000 Experimental study of the multipolar vortex instability. Phys. Rev. Lett. 85, 3400–3403.CrossRefGoogle ScholarPubMed
Eloy, C., Le Gal, P. & Le Dizès, S. 2003 Elliptic and triangular instabilities in rotating cylinders. J. Fluid Mech. 476, 357–388.CrossRefGoogle Scholar
Fukumoto, Y. 2003 The three dimensional instability of a strained vortex tube revisited. J. Fluid Mech. 493, 287–318.CrossRefGoogle Scholar
Gans, R. F. 1970 a On hydromagnetic precession in a cylinder. J. Fluid Mech. 45, 111.CrossRefGoogle Scholar
Gans, R. F. 1970 b On the precession of a resonant cylinder. J. Fluid Mech. 476, 865–872.CrossRefGoogle Scholar
Gans, R. F. 1984 Dynamics of a near-resonant fluid-filled gyroscope. AIAA J. 22, 1465–1471.CrossRefGoogle Scholar
Garg, S. C., Furunoto, N. & Vanyo, J. P. 1986 Spacecraft nutational instability prediction by energy dissipation measurments. J. Guid. 9 (3), 357–361.CrossRefGoogle Scholar
Greenspan, H. P. 1968 The Theory of Rotating Fluids. Cambridge University Press.Google Scholar
Hollerbach, R. & Kerswell, R. 1995 Oscillatory internal shear layers in rotating and precessing flows. J. Fluid Mech. 298, 327.CrossRefGoogle Scholar
Kelvin, L. 1880 Vibrations of a columnar vortex. Phil. Mag. 10, 155–168.Google Scholar
Kerswell, R. R. 1993 The instability of precessing flow. Geophys. Astrophys. Fluid Dyn. 72, 107–144.CrossRefGoogle Scholar
Kerswell, R. R. 1995 On the internal shear layers spawned by the critical regions in oscillatory Ekman boudary layers. J. Fluid Mech. 298, 311–325.CrossRefGoogle Scholar
Kerswell, R. R. 1996 Upper bounds on the energy dissipation in turbulent precession. J. Fluid Mech. 321, 335–370.CrossRefGoogle Scholar
Kerswell, R. R. 1999 Secondary instabilities in rapidly rotating fluids: inertial wave breakdown. J. Fluid Mech. 382, 283–306.CrossRefGoogle Scholar
Kerswell, R. R. 2002 Elliptical instability. Annu. Rev. Fluid Mech. 34, 83–113.CrossRefGoogle Scholar
Kerswell, R. R. & Barenghi, C. F. 1995 On the viscous decay rates of inertial waves in a rotating cylinder. J. Fluid Mech. 285, 203–214.CrossRefGoogle Scholar
Kobine, J. J. 1995 Inertial wave dynamics in a rotating and precessing cylinder. J. Fluid Mech. 303, 233–252.CrossRefGoogle Scholar
Kobine, J. J. 1996 Azimuthal flow associated with inertial wave resonance in a precessing cylinder. J. Fluid Mech. 319, 387–406.CrossRefGoogle Scholar
Kudlick, M. 1966 On the transient motions in a contained rotating fluid. PhD thesis, Massachussetts Institute of Technology, Cambridge, MA.Google Scholar
Lagrange, R., Eloy, C., Nadal, F. & Meunier, P. 2008 Instability of a fluid inside a precessing cylinder. Phys. Fluids 20 (8), 081701.CrossRefGoogle Scholar
Lambelin, J. P., Nadal, F., Lagrange, R. & Sarthou, A. 2009 Non-resonant viscous theory for the stability of a fluid-filled gyroscope. J. Fluid Mech. 639, 167–194.CrossRefGoogle Scholar
Lorenzani, S. & Tilgner, A. 2001 Fluid instabilities in precessing spheroidal cavities. J. Fluid Mech. 447, 111–128.CrossRefGoogle Scholar
Mahalov, A. 1993 The instability of rotating fluid columns subjected to a weak external coriolis-force. Phys. Fluids A 5 (4), 891–900.CrossRefGoogle Scholar
Malkus, W. V. R. 1968 Precession of the earth as the cause of geomagnetism. Science 160, 259–264.CrossRefGoogle ScholarPubMed
Malkus, W. V. R. 1989 An experimental study of global instabilities due to tidal (elliptical) distortion of a rotating elastic cylinder. Geophys. Astrophys. Fluid Dyn. 48, 123–134.CrossRefGoogle Scholar
Manasseh, R. 1992 Breakdown regimes of inertia waves in a precessing cylinder. J. Fluid Mech. 243, 261–296.CrossRefGoogle Scholar
Manasseh, R. 1996 Nonlinear behaviour of contained inertia waves. J. Fluid Mech. 315, 151–173.CrossRefGoogle Scholar
Mason, D. M. & Kerswell, R. R. 1999 Nonlinear evolution of the elliptical instability: an example of inertial breakdown. J. Fluid Mech. 396, 73–108.CrossRefGoogle Scholar
McEwan, A. D. 1970 Inertial oscillations in a rotating fluid cylinder. J. Fluid Mech. 40 (3), 603–640.CrossRefGoogle Scholar
Meunier, P., Eloy, C., Lagrange, R. & Nadal, F. 2008 A rotating fluid cylinder subject to weak precession. J. Fluid Mech. 599, 405–440.CrossRefGoogle Scholar
Meunier, P. & Leweke, T. 2003 Analysis and minimization of errors due to high gradients in particule image velocimetry. Exp. Fluids 35, 408–421.CrossRefGoogle Scholar
Moore, D. W. & Saffman, P. G. 1975 The instability of a straight vortex filament in a strain field. Proc. R. Soc. Lond. A 346, 413–425.Google Scholar
Noir, J. 2000 Écoulement d'un fluide dans une cavité en précession: approches numérique et expérimentale. PhD thesis, Université Joseph Fourier, Grenoble 1.Google Scholar
Noir, J., Brito, D., Aldridge, K. & Cardin, P. 2001 a Experimental evidence of inertial waves in a precessing spheroidal cavity. Geophys. Res. Lett. 38, 3785–3788.CrossRefGoogle Scholar
Noir, J., Cardin, P., Jault, D. & Masson, J. P. 2003 Experimental evidence of nonlinear resonance effects between retrograde precession and the tilt-over mode within a spheroid. Geophys. J. Intl 154, 407–416.CrossRefGoogle Scholar
Noir, J., Jault, D. & Cardin, P. 2001 b Numerical study of the motions within a slowly precessing sphere at low Ekman number. J. Fluid Mech. 437, 283–299.CrossRefGoogle Scholar
Poincaré, H. 1910 Sur la précession des corps déformables. Bull. Astro. 27, 257–264.Google Scholar
Racz, J.-P. & Scott, J. F. 2007 Parametric instability in a rotating cylinder of gas subject to sinusoidal axial compression. Part 2. Weakly nonlinear theory. J. Fluid Mech. 595, 291–321.CrossRefGoogle Scholar
Saffman, P. G. 1992 Vortex Dynamics. Cambridge University Press.Google Scholar
Stewartson, K. 1958 On the stability of a spinning top containing liquid. J. Fluid Mech. 5, 577–592.CrossRefGoogle Scholar
Thompson, R. 1970 Diurnal tides and shear instabiliies in a rotating cylinder. J. Fluid Mech. 40, 737–751.CrossRefGoogle Scholar
Tilgner, A. 1999 a Magnetohydrodynamic flow in precessing spherical shells. J. Fluid Mech. 379, 303–318.CrossRefGoogle Scholar
Tilgner, A. 1999 b Non-axisymmetric shear layers in precessing fluid ellipsoidal shells. Geophys. J. Intl 136, 629–636.CrossRefGoogle Scholar
Tilgner, A. 2005 Precession driven dynamos. Phys. Fluids 17, 034104.CrossRefGoogle Scholar
Tilgner, A. 2007 Kinematic dynamos with precession driven flow in a sphere. Geophys. Astrophys. Fluid Dyn. 101, 1–9.CrossRefGoogle Scholar
Tsai, C.-Y. & Widnall, S. E. 1976 The stability of short waves on a straight vortex filament in a weak externally imposed strain field. J. Fluid Mech. 73 (4), 721–733.CrossRefGoogle Scholar
Vanyo, J. P., Wilde, P. & Cardin, P. 1995 Experiments on precessing flows in the earth's liquid core. Geophys. J. Intl 121, 136–142.CrossRefGoogle Scholar
Waleffe, F. 1989 The 3d instability of a strained vortex and its relation to turbulence. PhD thesis, Massachusetts Institute of Technology, Cambridge, MA.Google Scholar
Waleffe, F. 1990 On the three-dimensional instability of strained vortices. Phys. Fluids A 2 (1), 76–80.CrossRefGoogle Scholar
Wu, C. & Roberts, P. 2008 A precessionally-driven dynamo in a plane layer. Geophys. Astrophys. Fluid Dyn. 102, 1–19.CrossRefGoogle Scholar
Wu, C. & Roberts, P. 2009 On a dynamo driven by topographic precession. Geophys. Astrophys. Fluid Dyn. 103, 467–501.CrossRefGoogle Scholar